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# 9.4 Indefinite Integrals

9.57   Theorem. Let be real numbers. If is a function that is integrable on each interval with endpoints in then

Proof: The case where is proved in theorem 8.18. The rest of the proof is exactly like the proof of exercise 5.69.

9.58   Exercise. Prove theorem 9.57.

We have proved the following formulas:

 (9.59) (9.60)

In each case we have a formula of the form

This is a general sort of situation, as is shown by the following theorem.

9.61   Theorem (Existence of indefinite integrals.) Let be an interval in , and let be a function such that is integrable on every subinterval of . Then there is a function such that for all

Proof: Choose a point and define

Then for any points in we have

We've used the fact that

9.62   Definition (Indefinite integral.) Let be a function that is integrable on every subinterval of an interval . An indefinite integral for on is any function such that for all .

A function that has an indefinite integral always has infinitely many indefinite integrals, since if is an indefinite integral for then so is for any number :

The following notation is used for indefinite integrals. One writes to denote an indefinite integral for . The here is a dummy variable and can be replaced by any available symbol. Thus, based on formulas (9.59) - (9.60), we write

We might also write

Some books always include an arbitrary constant with indefinite integrals, e.g.,

The notation for indefinite integrals is treacherous. If you see the two equations

and

then you want to conclude
 (9.63)

which is wrong. It would be more logical to let the symbol denote the set of all indefinite integrals for . If you see the statements

and

you are not tempted to make the conclusion in (9.63).

9.64   Theorem (Sum theorem for indefinite integrals) Let and be functions each of which is integrable on every subinterval of an interval , and let . Then
 (9.65)

Proof: The statement (9.65) means that if is an indefinite integral for and is an indefinite integral for , then is an indefinite integral for .

Let be an indefinite integral for and let be an indefinite integral for . Then for all

It follows that is an indefinite integral for .

9.66   Notation (.) If is a function defined on an interval , and if are points in we write for . The here is a dummy variable, and sometimes the notation is ambiguous, e.g. . In such cases we may write . Thus

while

Sometimes we write instead of .

9.67   Example. It follows from our notation that if is an indefinite integral for on an interval then

and this notation is used as follows:

In the last example I have implicitly used

9.68   Example. By using the trigonometric identities from theorem 9.21 we can calculate integrals of the form where are non-negative integers and . We will find

We have

so

Thus

Hence

The method here is clear, but a lot of writing is involved, and there are many opportunities to make errors. In practice I wouldn't do a calculation of this sort by hand. The Maple command
> int((sin(x))^3*cos(3*x),x=0..Pi/2);

responds with the value
                      - 5/12


9.69   Exercise. A Calculate the integrals

Then determine the values of

without doing any calculations. (But include an explanation of where your answer comes from.)

9.70   Exercise. Find the values of the following integrals. If the answer is geometrically clear then don't do any calculations, but explain why the answer is geometrically clear.

a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)

9.71   Exercise.

Arrange the numbers in increasing order. Try to do the problem without making any explicit calculations. By making rough sketches of the graphs you should be able to come up with the answers. Sketch the graphs, and explain how you arrived at your conclusion. No  proof" is needed.

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Ray Mayer 2007-09-07